New inequalities for polynomials
HTML articles powered by AMS MathViewer
- by C. Frappier, Q. I. Rahman and St. Ruscheweyh PDF
- Trans. Amer. Math. Soc. 288 (1985), 69-99 Request permission
Abstract:
Using a recently developed method to determine bound-preserving convolution operators in the unit disk, we derive various refinements and generalizations of the well-known inequalities of S. Bernstein and M. Riesz for polynomials. Many of these results take into account the size of one or more of the coefficients of the polynomial in question. Other results of similar nature are obtained from a new interpolation formula.References
- R. J. Duffin and A. C. Schaeffer, A refinement of an inequality of the brothers Markoff, Trans. Amer. Math. Soc. 50 (1941), 517–528. MR 5942, DOI 10.1090/S0002-9947-1941-0005942-4 W. L. Ferrar, Algebra, 2nd ed., Oxford Univ. Press, London, 1957.
- C. Frappier and Q. I. Rahman, On an inequality of S. Bernstein, Canadian J. Math. 34 (1982), no. 4, 932–944. MR 672687, DOI 10.4153/CJM-1982-066-7 F. R. Gantmacher, The theory of matrices, Chelsea, New York, 1959.
- A. Giroux and Q. I. Rahman, Inequalities for polynomials with a prescribed zero, Trans. Amer. Math. Soc. 193 (1974), 67–98. MR 352427, DOI 10.1090/S0002-9947-1974-0352427-3
- N. K. Govil, V. K. Jain, and G. Labelle, Inequalities for polynomials satisfying $p(z)\equiv z^{n}p(1/z)$, Proc. Amer. Math. Soc. 57 (1976), no. 2, 238–242. MR 414838, DOI 10.1090/S0002-9939-1976-0414838-4
- Michael Lachance, Edward B. Saff, and Richard S. Varga, Inequalities for polynomials with a prescribed zero, Math. Z. 168 (1979), no. 2, 105–116. MR 544699, DOI 10.1007/BF01214190
- M. A. Malik, On the derivative of a polynomial, J. London Math. Soc. (2) 1 (1969), 57–60. MR 249583, DOI 10.1112/jlms/s2-1.1.57
- Zeev Nehari, Conformal mapping, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1952. MR 0045823
- D. J. Newman, Polynomials and rational functions, Approximation theory and applications (Proc. Workshop, Technion—Israel Inst. Tech., Haifa, 1980) Academic Press, New York-London, 1981, pp. 265–282. MR 615416 G. Pólya and G. Szegö, Aufgaben und Lehrsatze aus der Analysis, Springer, Berliń, 1925.
- Qazi Ibadur Rahman, Applications of functional analysis to extremal problems for polynomials, Séminaire de Mathématiques Supérieures, No. 29 (Été, vol. 1967, Les Presses de l’Université de Montréal, Montreal, Que., 1968. MR 0251195
- Q. I. Rahman and G. Schmeisser, Some inequalities for polynomials with a prescribed zero, Trans. Amer. Math. Soc. 216 (1976), 91–103. MR 399427, DOI 10.1090/S0002-9947-1976-0399427-7
- Marcel Riesz, Über Einen Sat’z des Herrn Serge Bernstein, Acta Math. 40 (1916), no. 1, 337–347 (German). MR 1555142, DOI 10.1007/BF02418550
- W. Rogosinski and G. Szegö, Über die Abschnitte von Potenzreihen, die in einem Kreise beschränkt bleiben, Math. Z. 28 (1928), no. 1, 73–94 (German). MR 1544940, DOI 10.1007/BF01181146
- W. W. Rogosinski, Extremum problems for polynomials and trigonometrical polynomials, J. London Math. Soc. 29 (1954), 259–275. MR 62859, DOI 10.1112/jlms/s1-29.3.259
- Stephan Ruscheweyh, Convolutions in geometric function theory, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 83, Presses de l’Université de Montréal, Montreal, Que., 1982. Fundamental Theories of Physics. MR 674296
- A. C. Schaeffer, Inequalities of A. Markoff and S. Bernstein for polynomials and related functions, Bull. Amer. Math. Soc. 47 (1941), 565–579. MR 5163, DOI 10.1090/S0002-9904-1941-07510-5
- T. Sheil-Small, On the convolution of analytic functions, J. Reine Angew. Math. 258 (1973), 137–152. MR 320761, DOI 10.1515/crll.1973.258.137 G. Szegö, Über einen Satz des Herrn Serge Bernslein, Schriften Königsberger Gelehrten Gesellschaft 5 (1928), 59-70.
- M. Tsuji, Potential theory in modern function theory, Maruzen Co. Ltd., Tokyo, 1959. MR 0114894
- C. Visser, A simple proof of certain inequalities concerning polynomials, Nederl. Akad. Wetensch., Proc. 48 (1945), 276–281 = Indagationes Math. 7, 81–86 (1945). MR 15568
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 288 (1985), 69-99
- MSC: Primary 26D05; Secondary 30A10, 41A17
- DOI: https://doi.org/10.1090/S0002-9947-1985-0773048-1
- MathSciNet review: 773048